No, you are not automatically offered a cc, but it is easy enough to get one. And once you have a constant stream of income, very easy. The system clearly exploits people with bad urges.

You do not want to apply to too many credit card companies though, because each application creates and "inquiry" on your record and that counts against you. So it is at the same time good to have many cc, but applying for each additional one may have a negative effect, and you have to balance these two considerations in order to arrive at the best possible credit score for you. The same applies to asking for a credit line increase: if say Discover increases my credit line, my usage percentage or credit utilization ratio (there is an exact industry term for it but I am too lazy now to look it up) would fall which is beneficial for the credit score but the credit inquiry that the application would create is detrimental, and on balance I think it would be negative, so I do not apply for credit line increases.